The present disclosure relates, generally, to computer aided design systems and methods.
Finite element analysis (FEA) is a computational tool commonly used by engineers in designing parts and systems using computer aided design (CAD) software. FEA allows testing of mechanical properties of a computer designed part so that the part can be modified if it doesn't meet the required mechanical specifications. One of the limitations of FEA is that it requires use of an approximated version of the computer-modeled part. The approximated geometry tends to be blockier than the actual design, but this is required in order for conventional FEA to break the part into tiny elements, which are each analyzed. Curved surfaces do not easily break up into the types of elements needed for modern FEA. As a result the results are approximations of the performance of the actual part, which in some cases are pretty close, but which in other cases can be quite far off. These elements are connected together in the form of a mesh.
Isogeometric analysis is an analysis approach introduced by Hughes et al. in T. Hughes, J. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer methods in applied mechanics and engineering 194 (39) (2005) 4135-4195, which is incorporated herein by reference in its entirety. In isogeometric analysis, the same basis functions used to represent geometric models, such as Non-Uniform Rational B-Splines (NURBS), are also used to approximate field variables in solving partial differential equations (PDEs). Due to the same basis used in geometric representation and in solution approximation, isogeometric analysis eliminates the geometric approximation error commonly occurred in classical FEA procedures. Once an initial mesh is constructed, refinements can also be implemented and an exact geometry is maintained at all levels without the necessity of interaction with the CAD system. Another advantage of isogeometric analysis is its computational efficiency on a per-node basis over classical C0 Lagrange polynomial based finite element. The higher continuity of the NURBS basis has been demonstrated to significantly improve the numerical efficiency and accuracy on a per node basis in many areas including structural analysis, fluid simulation, and shape optimization.
Isogeometric analysis techniques relying on a basis other than NURBS have also been developed. To overcome the limitation of the tensor product structure of NURBS in local mesh refinement, methods based on subdivision solids and T-splines have been developed recently and have been successfully used in isogeometric analysis. The introduction of T-junction in T-splines allows T-splines to represent complex shapes in a single patch and permit local refinement. On the other hand, challenges exist with respect to analysis-suitable T-splines, such as how to obtain efficient local refinement and effective treatment of so-called extraordinary points.
Recently, triangular Bézier splines have emerged as a powerful alternative to shape modeling and isogeometric analysis due to their flexibility in representing shapes of complex topology and their higher order of continuity. Local refinement can also be implemented without any great difficulty. Various sets of basis functions have been defined on triangulations using bivariate spline functions. Some bivariate splines have also been effectively applied in solving PDEs, including some where quadratic Powell-Sabin (PS) splines with C1 smoothness are considered. A locally-supported basis is constructed by normalizing the piecewise quadratic PS B-splines and it is cast in the Bernstein-Bézier form. Such basis has global C1 smoothness and is used to approximate the solution of PDEs. More generalized Cr basis and elements including PS, Clough-Tocher (CT), and polynomial macro-elements based on rational triangular Bézier splines (rTBS) have also been introduced and applied successfully in isogeometric analysis on triangulations.
Referring to FIG. 1, cubic C1 smooth basis functions with CT macro-elements are illustrated within the context of a rTBS based isogeometric analysis. The given physical domain is triangulated into a set of C1 smooth Bézier elements, which are mapped from the parametric mesh. More particularly, free control points and domain points 10 are determined by dependent control points and domain points 12, respectively under the continuity constraints. Thus, the C1 basis functions ψ are constructed as linear combinations of the C0 Bernstein basis ϕ, under the continuity constraints. The resulting analysis has shown to be efficient, accurate, and convergent. However, optimal convergence in h-refinement has only been achieved for C0 elements and the convergence rate is sub-optimal for Cr elements.
Thus, it would be desirable to have a system and method for creating meshes from models with a controlled amount of and, preferably, without any approximation. Furthermore, it would be desirable that such system and method demonstrate effective and efficient convergence in the analysis.